Skip to main content
SpringerLink
Log in

On the Distributions of the State Sizes of Discrete Time Homogeneous Markov Systems

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

The evolution of a closed discrete-time homogeneous Markov system (HMS) is determined by the evolution of its state sizes in time. In order to examine the variability of the state sizes, their moments are evaluated for any time point, and recursive formulae for their computation are derived. As a consequence the asymptotic values of the moments for a convergent HMS can be evaluated. The respective recursive formula for a HMS with periodic transition matrix is given. The p.d.f.’s of the state sizes follow directly by means of the moments. The theoretical results are illustrated by a numerical example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • D. J. Bartholomew, Stochastic Models for Social Processes. 3rd edn, Wiley: New York, 1982.

    MATH  Google Scholar 

  • J. Gani, “Formulae for projecting enrolments and degrees awarded in universities,” Journal of the Royal Statistical Society. Series A. General vol. 126 pp. 400–409, 1963.

    Article  Google Scholar 

  • D. L. Isaacson and R. W. Madsen, Markov Chains. Theory and Applications, Wiley: New York, 1976.

    MATH  Google Scholar 

  • I. Kipouridis and G. Tsaklidis, “The size order of the state vector of discrete-time homogeneous Markov systems,” Journal of Applied Probability vol. 38 pp. 357–368, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  • S. I. McClean, B. McAlea, and P. Millard, “Using a Markov reward model to estimate spend-down costs for a geriatric department,” Journal of the Operational Research Society vol. 10 pp. 1021-1025, 1998.

    Article  Google Scholar 

  • P. D. Patoucheas and G. Stamou, “Non-homogeneous Markovian models in ecological modelling: a study of the zoobenthos dynamics in Thermaikos Gulf, Greece,” Ecological Modelling vol. 66 pp. 197–215, 1993.

    Article  Google Scholar 

  • G. J. Taylor, S. I. McClean, and P. Millard, “Stochastic model of geriatric patient bed occupancy behaviour,” Journal of the Royal Statistical Society vol. 163(1) pp. 39–48, 2000.

    Article  Google Scholar 

  • G. Tsaklidis, “The evolution of the attainable structures of a homogeneous Markov system with fixed size,” Journal of Applied Probability vol. 31 pp. 348–361, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  • G. Tsaklidis and K. P. Soldatos, “Modelling of continuous time homogeneous Markov system with fixed size as elastic solid,” Applied Mathematical Modelling vol. 27 pp. 877–887, 2003.

    Article  MATH  Google Scholar 

  • P.-C. G. Vassiliou, “Asymptotic behaviour of Markov systems,” Journal of Applied Probability vol. 19 pp. 851–857, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  • P.-C. G. Vassiliou, “The evolution of the theory of non-homogeneous Markov systems,” Applied Stochastic Models and Data Analysis vol. 13 no. 3–4 pp. 159–176, 1997.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

    G. Vasiliadis & G. Tsaklidis

Authors
  1. G. Vasiliadis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Tsaklidis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Tsaklidis.

Additional information

This research was partially supported by the State Scholarships Foundation of Greece.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vasiliadis, G., Tsaklidis, G. On the Distributions of the State Sizes of Discrete Time Homogeneous Markov Systems. Methodol Comput Appl Probab 10, 55–71 (2008). https://doi.org/10.1007/s11009-007-9034-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-007-9034-6

Keywords

AMS 2000 Subject Classification

Search

Navigation